Question

cpm8 : advanced algebra: concepts and connections - block (27.0831030)
modeling with rational functions
interpreting an intercept of a rational function that models a real - world situation
a horse owner has 50 lbs of hay that is 6% protein by weight. he adds ( x ) lbs of corn that is 12% protein by weight.
the function ( y=\frac{0.06(50)+0.12x}{50 + x} ) models the percent of protein, ( y ), in the final mixture of feed.
what is the ( y ) - intercept of the function that models the percent of protein in the final mixture?
the ( y ) - intercept is \\(\square\\)

Explanation:

Step1: Recall y-intercept definition

The y-intercept of a function is the value of \( y \) when \( x = 0 \). So we substitute \( x = 0 \) into the function \( y=\frac{0.06(50)+0.12x}{50 + x} \).

Step2: Substitute \( x = 0 \)

Substitute \( x = 0 \) into the function: \( y=\frac{0.06(50)+0.12(0)}{50+0} \).
First, calculate the numerator: \( 0.06(50)=3 \), and \( 0.12(0) = 0 \), so numerator is \( 3+0 = 3 \).
Denominator is \( 50+0 = 50 \).
Then \( y=\frac{3}{50}=0.06 \), which is \( 6\% \).

Answer:

6 (or 0.06, but since it's percent, 6 is appropriate as the model's y - intercept represents percent protein)